Staff: Mentor

One of the most important things in science is learning what the various equations represent and what the underlying assumptions are for that equation.

The time dilation equation and the length contraction equation are special cases, and do not always apply. The general formula is the Lorentz transform, and I strongly recommend that you use it instead of the shortcut formulas. Here is the Lorentz transform for one spatial dimension:
##t' = (t-vx/c^2)/\sqrt{1-v^2/c^2}##
##x' = (x-vt)/\sqrt{1-v^2/c^2}##

If you have the quantities ##\Delta t=t_b-t_a##, ##\Delta x=x_b-x_a##, ##\Delta t'=t'_b-t'_a##, and ##\Delta x'=x'_b-x'_a## then it is easy to show that:
##\Delta t' = (\Delta t-v\Delta x/c^2)/\sqrt{1-v^2/c^2}##
##\Delta x' = (\Delta x-v\Delta t)/\sqrt{1-v^2/c^2}##

If ##\Delta x = 0##, then you get the time dilation formula that you posted, however, in this problem ##\Delta x\ne 0##, so the time dilation formula does not apply. Similarly, the length contraction formula assumes that the object whose length is being measured is at rest in one of the frames, and since the runner is not at rest in either frame the length contraction formula also does not apply.